# Chapter 9 - Roots and Radicals - 9.3 - More on Simplifying Radicals - Problem Set 9.3: 54

$\dfrac{x\sqrt{y}}{3y^2}$

#### Work Step by Step

Using the properties of radicals, the given expression, $\dfrac{\sqrt{5x^2}}{\sqrt{45y^3}} ,$ simplifies to \begin{array}{l}\require{cancel} \sqrt{\dfrac{5x^2}{45y^3}} \\\\= \sqrt{\dfrac{\cancel{5}x^2}{\cancel{5}(9)y^3}} \\\\= \sqrt{\dfrac{x^2}{9y^3}} \\\\= \sqrt{\dfrac{x^2}{9y^2}\cdot\dfrac{1}{y}} \\\\= \sqrt{\left(\dfrac{x}{3y}\right)^2\cdot\dfrac{1}{y}} \\\\= \dfrac{x}{3y}\sqrt{\dfrac{1}{y}} \\\\= \dfrac{x}{3y}\sqrt{\dfrac{1}{y}\cdot\dfrac{y}{y}} \\\\= \dfrac{x}{3y}\sqrt{\dfrac{y}{y^2}} \\\\= \dfrac{x}{3y}\cdot\dfrac{\sqrt{y}}{\sqrt{y^2}} \\\\= \dfrac{x}{3y}\cdot\dfrac{\sqrt{y}}{y} \\\\= \dfrac{x\sqrt{y}}{3y^2} .\end{array} Note that all variables are assumed to represent positive real numbers.

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